Cryptology ePrint Archive: Report 2007/428

Isogenies and the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves

Benjamin Smith

Abstract: We describe the use of explicit isogenies
to reduce Discrete Logarithm Problems (DLPs)
on Jacobians of hyperelliptic genus~$3$ curves
to Jacobians of non-hyperelliptic genus~$3$ curves,
which are vulnerable to faster index calculus attacks.
We provide algorithms which compute an isogeny
with kernel isomorphic to $(\mathbb{Z}/2\mathbb{Z})^3$
for any hyperelliptic genus~$3$ curve.
These algorithms provide a rational isogeny
for a positive fraction of all hyperelliptic genus~$3$ curves
defined over a finite field of characteristic $p > 3$.
Subject to reasonable assumptions,
our algorithms provide an explicit and efficient
reduction from hyperelliptic DLPs to non-hyperelliptic DLPs
for around $18.57\%$ of all hyperelliptic genus~$3$ curves
over a given finite field.